My note refers
to a ‘key lemma’ and Anderson and Welty’s reply focuses on this key lemma
together with my arguments surrounding it. As this key lemma seems to be the
crux of our disagreement it is worthwhile setting out why I think it key.

“The Lord
of Non-Contradiction: An Argument for God from Logic” begins with the, largely,
uncontroversial claim that the laws of logic are necessarily true. By the end
of the paper we are invited to accept that the laws of logic are also
necessarily true thoughts. There is more content in the latter than the former,
content that can be argued to be contingent on God. My contention is that is
mainly the claim

“(s)ince
[the laws of logic] are true in every possible world, they must exist in every
possible world.” (Anderson & Welty, 2011)

that
allows the introduction of the excess content. Thus, it is key. (But it is not
the argument, it is a step on the way, a lemma; if you will a “staging post”.)

Existence
can be taken to imply possession of properties, natures, essential
natures and the like together with various assumptions about what else is
needed for these properties to subsist in. Or "being true" can
be taken as a property in itself, entirely sufficient to establish existence of
the laws of logic whether or not any other properties are held; in particular
whether or not they are thought.

This is the
equivocation: before the lemma discussion is limited to being true
whether or not thought, after the lemma discussion surrounds a concept of "exists" that requires thought.
Thought then entails a thinker, necessary thought a necessary thinker and then to
God. On the “left” we have “thought”, on
the right we have “truth and thought”.
My note explored possible ways of making the link (“if thought and truth
then thought” is valid, “if truth then truth and thought” is not). Unfortunately the valid destroy Anderson and
Welty’s argument whilst those that support it are invalid. This is as we should
expect:

“That the
laws of logic are necessarily true entails that they are true whether or not
God exists.”(Lloyd, 2012)

Anderson
and Welty, in their reply, claim that I offered no argument for that claim. One
seems, frankly, entirely redundant. “Necessary” is “not contingent” and “P is
not contingent on Q” is equivalent to “there does not exist a Q such that P is
contingent on Q”: there is no need to list all the infinite non-existent Qs
that P is contingent on or the existent Qs that P is not contingent on.

A
reference to “impossible worlds” is then made charging that “necessary” does
not entail existence in impossible
worlds. I fail to see the point of this objection. Do they mean to suggest that
a world without God is an impossible world? If so then (if the laws of logic
are necessary) there is no possible world where God does not exist and the laws
of logic do not exist. In a footnote,
presumably intended to be explanatory, they invite comparison with “(i)f the
proposition God exists is necessarily true then it is true whether or not God
exists” (Anderson &
Welty, 2013)
as if there is something problematical with this statement. What does “P
whether or not Q” mean? Its logical meaning is, simply, “P”. If we wish to emphasise the non dependence on
Q: “if Q then P and if not Q then P”. Anderson and Welty’s “problematic”
comparator, then, can be written:

□G
entails □ (G®G) and □ (¬G®G)

"G ®G” ,
together with "G", entails "G". “¬G ®G” is
just a long-winded way of saying “G”, as the truth table shows:

So the statement
simplifies to "□G".

A further
claim is that I “presuppose(d) that the laws of logic are not
ontologically dependent on God”. There was, of course, no
“presupposition”:“that P entails Q” is a statement of logical and linguistic
analysis that takes no position, presupposed or otherwise, on the truth either
of P or of Q.

(A reader
interested in a proof of "that P is necessary entails P whether or not
Q" is invited to consult the box at the foot of this post.)

At the
end of my note I sketched out the idea that all arguments to God from logic are
liable to fail. Naturally, anyone arguing that logic depends on God needs to
argue that logic is contingent on God. There are plenty of people who are quite
happy to accept logic's contingency, but on something much more mundane than
God. Perhaps Paula thinks that the laws of logic are thought. She is quite happy that they exist because
there are beings, at least in this world, where there are minds, like Paula’s,
that think them. Paula’s position is reasonable and the God-from-logic
proponent requires necessity (having admitted contingency) in order to
discomfit Paula’s position. The
God-from-logic proponent needs, so to speak, to place the laws logic in a world
where Paula is not there to think them.
The laws of logic must be there, because they are necessary, but can’t (according
to Paula) because they are contingent on her. The assumption of necessity per se though is not enough to save the
argument. Necessity removes the
contingency and without contingency the laws of logic cannot be contingent on
God. So the laws of logic must be held both necessary and contingent.

Now P can
be both necessary and contingent if the necessity is qualified. A proper subset
of possible worlds is those possible worlds where the physical laws of our
universe hold. The laws of physics could have been different, so there are
possible worlds were they are different.
But we may limit our considerations to those possible worlds where the
laws of physics do hold. Something, such as the speed of light in a vacuum,
which holds in all of these physically possible worlds, is physically necessary. As the
speed of light could have been different there is at least one ‘metaphysical’ world
where it is. Thus the speed of light in
a vacuum is physically necessary and metaphysically contingent.

I do not
think this helps the God-from-logic proponent. It rescues him from
contradiction, but also rescues Paula. She,
herself, may adopt the position that the necessity of the laws of logic she has
been presented with is a qualified necessity.
She may admit that the laws of logic are X-ly necessary and still
maintain that they are Paula’s- mind-ly contingent.

There is
much to do, should anyone wish to pursue it, in analysing the interrelations of
qualified necessity. On first sight it
would seem that metaphysical contingencies can be physically necessary but metaphysical
necessities cannot be physical contingencies.
What of other ways of qualifying necessity; ontological, logical,
epistemological and the like?

I suspect
that no argument for God from logic will succeed mostly because I suspect that
no combination of qualifications will place Paula in a bind and not give her
the very tools to free herself.

There is
also the feeling that logical necessity is the nearest a qualified necessity
gets to necessity simpliciter. And logical necessity appears to take a
special role in the argument. Take
Anderson and Welty’s own summary of the qualification of the contingencies in
their argument:

"The
laws of logic are “contingent on God” only in the sense that they are
metaphysically dependent on God’s existence, in precisely the way that God’s
thoughts are metaphysically dependent on God’s existence. This doesn’t entail
that the laws of logic exist contingently or are true contingently (where
contingently is a modal operator equivalent to not necessarily)."(Anderson &
Welty, 2013)

Note, in
particular the status of truth. The
metaphysical contingency of the laws of logic does not mean that they are not
necessarily true. That does not sit well
with:

“(O)ne
can logically argue against God only if God exists”(Anderson &
Welty, 2011)

Just what
are Anderson and Welty to say to someone who argues against God using logic? “Yes,
yes, it may be truethat Goddoes not exist but part of the
logic you use to conclude that is ontologically suspect”?

Proof that □P entails □(Q®P) & □ (¬Q®P)

1

□P

Premise

2

◊¬ (Q®P) v ◊¬
(¬Q®P)

Negation
of the conclusion: “possibly not (Q®P) or
possibly not (¬Q®P)”

4

0r1

To test
the first option we need to open up a new possible world. As (by hypothesis)
“possibly not (Q ® P)”
there must be a world that has ¬(Q ® P)

5

¬(Q ® P)

And
here it is.

6

Q

5 is
true (and thus (Q ® P)
false) just when Q is true and...

7

¬P

P is
false

8

P

But
from our premise, P is in this world

X

Lines 7
and 8 contradict

9

0r2

Open up
another world. As “¬(¬Q ® P)” is
(by hypothesis) possible there must be a world that has ¬(¬Q ®
P).This need not be the same world as before, so a new one is needed.